I want to close gaps in a model to make it printable
The problem seems to be that you have way too many vertices it would be expected that the plane makes many triangle faces and looks very buggy.
A way to approach this is by not using a plane at all. Try hiding everything that is not the underside of the long coat (bottom of the torso if you will). Then you can switch to bottom view by pressing Ctrl + Numpad 7 and box selecting (B) all the faces you see. Afterwards you can extrude these faces down and flatten them on the Z axis with S + Z + 0. This should create a flat surface that you can drag to a desired height.
P.S. If you don't have vertex groups and are finding hiding everything difficult, you can also circle select (C) the underside faces manually.